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I need help with the expansion of this integral:

[tex]\int_0^\infty Z(x) J_o(\lambda x)dx[/tex] for [tex]\lambda \rightarrow \infty[/tex]

where I know that [tex]Z(x)\sim x^\sqrt{2}[/tex] for small [tex]x[/tex] and

exponentially small for large [tex]x[/tex]

It seems with other examples that I have done that the major contribution to the integral comes from the region [tex]x\sim 1/\lambda[/tex]. For larger [tex]x[/tex] the integrand oscillates rapidly and the integration cancels. One change of variable (re-scaling) that you may try is [tex]t=\lambda x[/tex]. But if you do it you end up with a divergent integral. And at first glance the original integral is convergent. Any hints?

Thanks.

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# Asymptotic expansion integral

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